schurs partition theorem

Input interpretation

Schur's partition theorem

Alternate name

Schur's theorem

Definition

Schur's partition theorem lets A(n) denote the number of partitions of n into parts congruent to ± 1 (mod 6), B(n) denote the number of partitions of n into distinct parts congruent to ± 1 (mod 3), and C(n) the number of partitions of n into parts that differ by at least 3, with the added constraint that the difference between multiples of three is at least 6. Then A(n) = B(n) = C(n). The values of A(n) = B(n) = C(n) for n = 1, 2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (OEIS A003105). For example, for n = 15, there are nine partitions satisfying these conditions, as summarized in the following table.
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Related terms

Andrews-Gordon identity | Göllnitz's theorem | Rogers-Ramanujan identities | Schur number | Schur's lemma
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